## Lecture Notes: Introduction to Geometric Measure Theory

Reference: Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, by Francesco Maggi.

### Lecture Notes

#### Part I:

Lecture 1: Outer measures, measure theory and integration.

Lecture 2: Borel and Radon measures.

Lecture 3: Hausdorff measures, dimension, isodiametric inequality.

Lecture 4: Radon measures and continuous functions.

Lecture 5: Besicovitch's covering theorem, differentiation of Radon measures.

Lecture 6: Campanato's criterion.

Lecture 7: Lower dimensional measures of Radon measures, Rademacher's theorem.

Lecture 8: Rectifiable sets I.

Lecture 9: Rectifiable sets II.

Lecture 10: Rectifiable sets III.

Lecture 11: Lipschitz linearization. Area formula.

Lecture 12: Sets of finite perimeter.

Lecture 13: Compactness of sets of finite perimeter.

Lecture 14: Existence of minimizers in geometric variational problems.

Lecture 15: Coarea formula, approximation of sets of finite perimeter.

Lecture 16: The Euclidean isoperimetric problem I.

Lecture 17: The Euclidean isoperimetric problem II.

Lecture 18: Reduced boundary. Tangential properties of the reduced boundary.

Lecture 19: The reduced boundary is locally (n-1)-rectifiable. Federer's theorem.

Lecture 20: Proof of De Giorgi's structure theorem using Whitney extension theorem.

#### Part II (Regularity of minimizers)

Lecture 21: Comparison sets. Density estimates for perimeter minimizers.

Lecture 22: First variation of perimeter.

Lecture 23: Stationary sets and monotonicity of density ratios.

Lecture 24: Area and coarea formula for rectifiable sets. Monotonicity revisited.

Lecture 25: The Lipschitz graph criterion. The area functional and the minimal surface equation.

Lecture 26: Compacteness for sequences of minimizers. Basic properties of the excess.

Lecture 27: Lower semicontinuity of the excess. Small excess position. Excess measure.

Lecture 28: The height bound theorem.

Lecture 29: The Lipschitz approximation theorem I.

Lecture 30: The Lipschitz approximation theorem II.

Lecture 31: The reverse Poincare inequality. Two lemmas on harmonic functions.

Lecture 32: The "excess improvement by tilting" estimate.

Lecture 33: Lipschitz continuity of local perimeter minimizers.

Lecture 34: C^{1,gamma} regularity of local perimeter minimizers.

Lecture 35: Higher regularity.

Lecture 36: Analysis of singularities I: Monotonicity formula.

Lecture 37: Analysis of singularities II. Simons' theorem.

Lecture 38: Analysis of singularities III. Federer's dimension reduction argument.